Sorinel A. Oprisan

Modeling Pharmacological Clock and Memory Patterns of Interval Timing in a Striatal Beat-Frequency Model with Realistic, Noisy Neurons

In most species, the capability of perceiving and using the passage of time in the seconds-to-minutes range (interval timing) is not only accurate but also scalar: errors in time estimation are linearly related to the estimated duration. The ubiquity of scalar timing extends over behavioral, lesion, and pharmacological manipulations. For example, in mammals, dopaminergic drugs induce an immediate, scalar change in the perceived time (clock pattern), whereas cholinergic drugs induce a gradual, scalar change in perceived time (memory pattern). How do these properties emerge from unreliable, noisy neurons firing in the milliseconds range? Neurobiological information relative to the brain circuits involved in interval timing provide support for an striatal beat frequency (SBF) model, in which time is coded by the coincidental activation of striatal spiny neurons by cortical neural oscillators. While biologically plausible, the impracticality of perfect oscillators, or their lack thereof, questions this mechanism in a brain with noisy neurons. We explored the computational mechanisms required for the clock and memory patterns in an SBF model with biophysically realistic and noisy Morris–Lecar neurons (SBF–ML). Under the assumption that dopaminergic drugs modulate the firing frequency of cortical oscillators, and that cholinergic drugs modulate the memory representation of the criterion time, we show that our SBF–ML model can reproduce the pharmacological clock and memory patterns observed in the literature. Numerical results also indicate that parameter variability (noise) – which is ubiquitous in the form of small fluctuations in the intrinsic frequencies of neural oscillators within and between trials, and in the errors in recording/retrieving stored information related to criterion time – seems to be critical for the time-scale invariance of the clock and memory patterns.

Time-scale invariance as an emergent property in a perceptron with realistic, noisy neurons

In most species, interval timing is time-scale invariant: errors in time estimation scale up linearly with the estimated duration. In mammals, time-scale invariance is ubiquitous over behavioral, lesion, and pharmacological manipulations. For example, dopaminergic drugs induce an immediate, whereas cholinergic drugs induce a gradual, scalar change in timing. Behavioral theories posit that time-scale invariance derives from particular computations, rules, or coding schemes. In contrast, we discuss a simple neural circuit, the perceptron, whose output neurons fire in a clockwise fashion based on the pattern of coincidental activation of its input neurons. We show numerically that time-scale invariance emerges spontaneously in a perceptron with realistic neurons, in the presence of noise. Under the assumption that dopaminergic drugs modulate the firing of input neurons, and that cholinergic drugs modulate the memory representation of the criterion time, we show that a perceptron with realistic neurons reproduces the pharmacological clock and memory patterns, and their time-scale invariance, in the presence of noise. These results suggest that rather than being a signature of higher order cognitive processes or specific computations related to timing, time-scale invariance may spontaneously emerge in a massively connected brain from the intrinsic noise of neurons and circuits, thus providing the simplest explanation for the ubiquity of scale invariance of interval timing.

The influence of limit cycle topology on the phase resetting curve

Understanding the phenomenology of phase resetting is an essential step toward developing a formalism for the analysis of circuits composed of bursting neurons that receive multiple, and sometimes overlapping, inputs. If we are to use phase-resetting methods to analyze these circuits, we can either generate phase-resetting curves (PRCs) for all possible inputs and combinations of inputs, or we can develop an understanding of how to construct PRCs for arbitrary perturbations of a given neuron. The latter strategy is the goal of this study. We present a geometrical derivation of phase resetting of neural limit cycle oscillators in response to short current pulses. A geometrical phase is defined as the distance traveled along the limit cycle in the appropriate phase space. The perturbations in current are treated as displacements in the direction corresponding to membrane voltage. We show that for type I oscillators, the direction of a perturbation in current is nearly tangent to the limit cycle; hence, the projection of the displacement in voltage onto the limit cycle is sufficient to give the geometrical phase resetting. In order to obtain the phase resetting in terms of elapsed time or temporal phase, a mapping between geometrical and temporal phase is obtained empirically and used to make the conversion. This mapping is shown to be an invariant of the dynamics. Perturbations in current applied to type II oscillators produce significant normal displacements from the limit cycle, so the difference in angular velocity at displaced points compared to the angular velocity on the limit cycle must be taken into account. Empirical attempts to correct for differences in angular velocity (amplitude versus phase effects in terms of a circular coordinate system) during relaxation back to the limit cycle achieved some success in the construction of phase-resetting curves for type II model oscillators. The ultimate goal of this work is the extension of these techniques to biological circuits comprising type II neural oscillators, which appear frequently in identified central pattern-generating circuits.

Prediction of Entrainment And 1:1 Phase-Locked Modes in Two-Neuron Networks Based on the Phase Resetting Curve Method

This study elaborated a systematic and consistent technique for predicting 1:1 entrainment patterns in two-neuron networks based on the phase resetting curve (PRC). The graphical method of the first-order PRC is intuitive and successfully predicts both the entrained period and the phase difference between neural oscillators. It was found that the shape of the synaptic current is not critical in determining the PRC, but the area of the injected current versus time, which is the exchanged electric charge, is a crucial factor. The PRC-based existence and stability criterion for entrainment is in good agreement with the experimental data.

A Geometric Approach to Phase Resetting Estimation Based on Mapping Temporal to Geometric Phase

The membrane potential is the most commonly traced quantity in both numerical simulations and electrophysiological experiments. One quantitative measure of neuronal activity that could be extracted from membrane potential is the firing period. The phase resetting curve (PRC) is a quantitative measure of the relative change in the firing period of a neuron due to external perturbations such as synaptic inputs. The experimentally recorded periodic oscillations of membrane potential represent a one-dimensional projection of a closed trajectory, or limit cycle, in neuron’s multidimensional phase space. This chapter is entirely dedicated to the study of the relationship between the PRC and the geometry of the phase space trajectory. This chapter focuses on systematically deriving the mappings σ=σ(φ, μ) between the temporal phase φ and the geometric phase σ when some parameters μ are perturbed. For this purpose, both analytical approaches, based on the vector fields of a known theoretical models, and numerical approaches, based on experimentally recorded membrane potential, are discussed in the context of phase space reconstruction of limit cycle. The natural reference frame attached to neuron’s unperturbed limit cycle, γ breaks the perturbation of control parameter μ into tangent and normal displacements relative to the unperturbed γ. Detailed derivations of PRC in response to weak tangent and normal, perturbations of γ are provided. According to the geometric approach to PRC prediction, a hard, external perturbation forces the figurative point to cross the excitability threshold, or separatrix, in the phase space. The geometric method for PRC prediction detailed in this chapter gives accurate predictions both for hard inhibitory and excitatory perturbations of γ. The geometric method was also successfully generalized to a more realistic case of a neuron receiving multiple inputs per cycle.

Experimental study of nonequilibrium fluctuations during free diffusion in nanocolloids using microscopic techniques

We report quantitative experimental results regarding concentration fluctuations based on a small-angle light-scattering setup. A shadowgraph technique was used to record concentration fluctuations in a free-diffusion cell filled with colloids. Our experimental setup includes an objective attached to the CCD camera to increase the field of view. We performed two separate experiments, one with 20 nm gold and the other with 200 nm silica colloids, and extracted both the structure factors and the correlation time during the early stages of concentration fluctuations. The temporal evolution of fluctuations was also qualitatively investigated using recursive plots and spatial-temporal sections of fluctuating images. We found that the correlation time versus wavenumber for gold nanocolloids is concave shaped, whereas, for silica colloids, it is convex shaped. The difference in correlation time behavior is not only due to the size of the particle, but also to possible plasmonic interactions in gold colloids.

Phase resetting and its implications for interval timing with intruders

Time perception in the second-to-minutes range is crucial for fundamental cognitive processes like decision making, rate calculation, and planning. We used a striatal beat frequency (SBF) computational model to predict the response of an interval timing network to intruders, such as gaps in conditioning stimulus (CS), or distracters presented during the uninterrupted CS. We found that, depending on the strength of the input provided to neural oscillators by the intruder, the SBF model can either ignore it or reset timing. The significant delays in timing produced by emotionally charged distracters were numerically simulated by a strong phase resetting of all neural oscillators involved in the SBF network for the entire duration of the evoked response. The combined effect of emotional distracter and pharmacological manipulations was modeled in our SBF model by modulating the firing frequencies of neural oscillators after they are released from inhibition due to emotional distracters. This article is part of a Special Issue entitled: SI: Associative and Temporal Learning.

Clocks within clocks: timing by coincidence detection

The many existent models of timing rely on vastly different mechanisms to track temporal information. Here we examine these differences, and identify coincidence detection in its most general form as a common mechanism that many apparently different timing models share, as well as a common mechanism of biological circadian, millisecond and interval timing. This view predicts that timing by coincidence detection is a ubiquitous phenomenon at many biological levels, explains the reports of biological timing in many brain areas, explains the role of neural noise at different time scales at both biological and theoretical levels, and provides cohesion within the timing field.

Why noise is useful in functional and neural mechanisms of interval timing

BackgroundThe ability to estimate durations in the seconds-to-minutes range - interval timing - is essential for survival, adaptation and its impairment leads to severe cognitive and/or motor dysfunctions. The response rate near a memorized duration has a Gaussian shape centered on the to-be-timed interval (criterion time). The width of the Gaussian-like distribution of responses increases linearly with the criterion time, i.e., interval timing obeys the scalar property.ResultsWe presented analytical and numerical results based on the striatal beat frequency (SBF) model showing that parameter variability (noise) mimics behavioral data. A key functional block of the SBF model is the set of oscillators that provide the time base for the entire timing network. The implementation of the oscillators block as simplified phase (cosine) oscillators has the additional advantage that is analytically tractable. We also checked numerically that the scalar property emerges in the presence of memory variability by using biophysically realistic Morris-Lecar oscillators. First, we predicted analytically and tested numerically that in a noise-free SBF model the output function could be approximated by a Gaussian. However, in a noise-free SBF model the width of the Gaussian envelope is independent of the criterion time, which violates the scalar property. We showed analytically and verified numerically that small fluctuations of the memorized criterion time leads to scalar property of interval timing.ConclusionsNoise is ubiquitous in the form of small fluctuations of intrinsic frequencies of the neural oscillators, the errors in recording/retrieving stored information related to criterion time, fluctuation in neurotransmitters’ concentration, etc. Our model suggests that the biological noise plays an essential functional role in the SBF interval timing.

Existence and stability criteria for phase-locked modes in ring neural networks based on the spike time resetting curve method

We developed a systematic and consistent mathematical approach to predicting 1:1 phase-locked modes in ring neural networks of spiking neurons based on the open loop spike time resetting curve (STRC) and its almost equivalent counterpart-the phase resetting curve (PRC). The open loop STRCs/PRCs were obtained by injecting into an isolated model neuron a triangular shaped time-dependent stimulus current closely resembling an actual synaptic input. Among other advantages, the STRC eliminates the confusion regarding the undefined phase for stimuli driving the neuron outside of the unperturbed limit cycle. We derived both open loop PRC and STRC-based existence and stability criteria for 1:1 phase-locked modes developed in ring networks of spiking neurons. Our predictions were in good agreement with the closed loop numerical simulations. Intuitive graphical methods for predicting phase-locked modes were also developed both for half-centers and for larger ring networks.